We consider a class of coupled systems with damping terms. By using multiplier method and the estimation techniques of the energy, we show that even if the kernel function is nonincreasing and integrable without additional conditions, the energy of the system decays also to zero in a good rate.
Kun-Peng Jin. "Stability of a Class of Coupled Systems." Abstr. Appl. Anal. 2014 (SI13) 1 - 5, 2014. https://doi.org/10.1155/2014/835765